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Whenever you see a card's own card name within the the rules text of a card, it isn't referring to all cards with that name. It is referring to itself. I believe at one point, the Oracle text replaced this self reference with ~this~ (or perhaps, the database allowed searching using "this"). You can think of the triggered ability in that way:

When [~this~] dies, you may have it deal 1 damage to target creature or player.

From the Comprehensive Rules:

201.4. Text that refers to the object it's on by name means just that particular object and not any other objects with that name, regardless of any name changes caused by game effects.

If you were looking for abilities that would trigger multiple times, you need cards that trigger "Whenever a Goblin ...". These cards only care about the creature subtype in question. For example:

Boggart Mob - Whenever a Goblin you control deals combat damage to a player, you may put a 1/1 black Goblin Rogue creature token onto the battlefield.

Boggart Shenanigans- Whenever another Goblin you control dies, you may have Boggart Shenanigans deal 1 damage to target player.

Two Shenanigans won't trigger off of themselves, because of the another, but both will trigger when another Goblin permanent is sent to the graveyard from the battlefield. (Note: Boggart Shenanigans will do 1 damage twice, which is not the same thing as to deal 2 damage once.)

I suppose the templating for a Goblin Arsonist that would double its ability would begin something like: "Whenever a creature called Goblin Arsonist dies..."
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thesunneversetsAug 14 '12 at 10:20

@thesunneversets, if we mean double in the OPs sense, not the FoR sense, then something like the Dwarven Shrine template mixed with Zubera's and the Burst cards, "When ~this~ dies, it deals X damage to target creature or player where X is one plus the number of creatures with the same name."
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user1873Aug 14 '12 at 13:07

Aha, I see what you mean, I didn't get the idea that the OP cared about whether the 2 damage was dealt as one event or two! But I fully accept that it's more fun to be extra-pedantic about Magic questions :)
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thesunneversetsAug 14 '12 at 19:19